# Exercises: Fractions (1.4)

## Exercises: Find Equivalent Fractions

Instructions: For questions 1-4, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

1. $\frac{3}{8}$

Solution

$\frac{6}{16},\frac{9}{24},\frac{12}{32}$ (Answers may vary)

2. $\frac{5}{8}$

3. $\frac{5}{9}$

Solution

$\frac{10}{18},\frac{15}{27},\frac{20}{36}$ (Answers may vary)

4. $\frac{1}{8}$

## Exercises: Simplify Fractions

Instructions: For questions 5-14, simplify.

5. $-\frac{40}{88}$

Solution

$-\frac{5}{11}$

6. $-\frac{63}{99}$

7. $-\frac{108}{63}$

Solution

$-\frac{12}{7}$

8. $-\frac{104}{48}$

9. $\frac{120}{252}$

Solution

$\frac{10}{21}$

10. $\frac{182}{294}$

11. $-\frac{3x}{12y}$

Solution

$-\frac{x}{4y}$

12. $-\frac{4x}{32y}$

13. $\frac{14{x}^{2}}{21y}$

Solution

$\frac{2{x}^{2}}{3y}$

14. $\frac{24a}{32{b}^{2}}$

## Exercises: Multiply Fractions

Instructions: For questions 15-30, multiply.

15. $\frac{3}{4}\cdot\frac{9}{10}$

Solution

$\frac{27}{40}$

16. $\frac{4}{5}\cdot\frac{2}{7}$

17. $-\frac{2}{3}\left(-\frac{3}{8}\right)$

Solution

$\frac{1}{4}$

18. $-\frac{3}{4}\left(-\frac{4}{9}\right)$

19. $-\frac{5}{9}\cdot\frac{3}{10}$

Solution

$-\frac{1}{6}$

20. $-\frac{3}{8}\cdot\frac{4}{15}$

21. $\left(-\frac{14}{15}\right)\left(\frac{9}{20}\right)$

Solution

$-\frac{21}{50}$

22. $\left(-\frac{9}{10}\right)\left(\frac{25}{33}\right)$

23. $\left(-\frac{63}{84}\right)\left(-\frac{44}{90}\right)$

Solution

$\frac{11}{30}$

24. $\left(-\frac{63}{60}\right)\left(-\frac{40}{88}\right)$

25. $4\cdot\frac{5}{11}$

Solution

$\frac{20}{11}$

26. $5\cdot\frac{8}{3}$

27. $\frac{3}{7}\cdot21n$

Solution

$9n$

28. $\frac{5}{6}\cdot30m$

29. $(-8)\left(\frac{17}{4}\right)$

Solution

$-34$

30. $(-1)\left(-\frac{6}{7}\right)$

## Exercises: Divide Fractions

Instructions: For questions 31-44, divide.

31. $\frac{3}{4}\div\frac{2}{3}$

Solution

$\frac{9}{8}$

32. $\frac{4}{5}\div\frac{3}{4}$

33. $-\frac{7}{9}\div\left(-\frac{7}{4}\right)$

Solution

$1$

34. $-\frac{5}{6}\div\left(-\frac{5}{6}\right)$

35. $\frac{3}{4}\div\frac{x}{11}$

Solution

$\frac{33}{4x}$

36. $\frac{2}{5}\div\frac{y}{9}$

37. $\frac{5}{18}\div\left(-\frac{15}{24}\right)$

Solution

$-\frac{4}{9}$

38. $\frac{7}{18}\div\left(-\frac{14}{27}\right)$

39. $\frac{8u}{15}\div\frac{12v}{25}$

Solution

$\frac{10u}{9v}$

40. $\frac{12r}{25}\div\frac{18s}{35}$

41. $-5\div\frac{1}{2}$

Solution

$-10$

42. $-3\div\frac{1}{4}$

43. $\frac{3}{4}\div\left(-12\right)$

Solution

$-\frac{1}{16}$

44. $-15\div\left(-\frac{5}{3}\right)$

## Exercises: Simplify by Dividing

Instructions: For questions 45-50, simplify.

45. $\displaystyle\frac{-\frac{8}{21}}{\frac{12}{35}}$

Solution

$-\frac{10}{9}$

46. $\displaystyle\frac{-\frac{9}{16}}{\frac{33}{40}}$

47. $\displaystyle\frac{-\frac{4}{5}}{2}$

Solution

$-\frac{2}{5}$

48. $\displaystyle\frac{5}{\frac{3}{10}}$

49. $\displaystyle\frac{\frac{m}{3}}{\frac{n}{2}}$

Solution

$\frac{2m}{3n}$

50. $\displaystyle\frac{-\frac{3}{8}}{-\frac{y}{12}}$

## Exercises: Simplify Expressions Written with a Fraction Bar

Instructions: For questions 51-70, simplify.

51. $\frac{22+3}{10}$

Solution

$\frac{5}{2}$

52. $\frac{19-4}{6}$

53. $\frac{48}{24-15}$

Solution

$\frac{16}{3}$

54. $\frac{46}{4+4}$

55. $\frac{-6+6}{8+4}$

Solution

$0$

56. $\frac{-6+3}{17-8}$

57. $\frac{4\cdot3}{6\cdot6}$

Solution

$\frac{1}{3}$

58. $\frac{6\cdot6}{9\cdot2}$

59. $\frac{{4}^{2}-1}{25}$

Solution

$\frac{3}{5}$

60. $\frac{{7}^{2}+1}{60}$

61. $\frac{8\cdot3+2\cdot9}{14+3}$

Solution

$2\frac{8}{17}$

62. $\frac{9\cdot6-4\cdot7}{22+3}$

63. $\frac{5\cdot6-3\cdot4}{4\cdot5-2\cdot3}$

Solution

$\frac{3}{5}$

64. $\frac{8\cdot9-7\cdot6}{5\cdot6-9\cdot2}$

65. $\frac{{5}^{2}-{3}^{2}}{3-5}$

Solution

$-8$

66. $\frac{{6}^{2}-{4}^{2}}{4-6}$

67. $\frac{7\cdot4-2(8-5)}{9\cdot3-3\cdot5}$

Solution

$\frac{11}{6}$

68. $\frac{9\cdot7-3(12-8)}{8\cdot7-6\cdot6}$

69. $\frac{9(8-2)-3(15-7)}{6(7-1)-3(17-9)}$

Solution

$\frac{5}{2}$

70. $\frac{8(9-2)-4(14-9)}{7(8-3)-3(16-9)}$

## Exercises: Translate Phrases to Expressions with Fractions

Instructions: For questions 71-74, translate each English phrase into an algebraic expression.

71. the quotient of $r$ and the sum of $s$ and $10$

Solution

$\frac{r}{s+10}$

72. the quotient of $A$ and the difference of $3$ and $B$

73. the quotient of the difference of $x$ and $y$, and $-3$

Solution

$\frac{x-y}{-3}$

74. the quotient of the sum of $m$ and $n$, and $4q$

## Exercises: Everyday Math

Instructions: For questions 75-78, answer the given everyday math word problems.

75. Baking. A recipe for chocolate chip cookies calls for $\frac{3}{4}$ cup brown sugar. Imelda wants to double the recipe.

a. How much brown sugar will Imelda need? Show your calculation.
b. Measuring cups usually come in sets of $\frac{1}{4},\frac{1}{3},\frac{1}{2},\text{ and }1$ cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the cookie recipe.

Solution

a.$1\frac{1}{2}$ cups
b. Answers will vary

76. Baking. Nina is making $4$ pans of fudge to serve after a music recital. For each pan, she needs $\frac{2}{3}$ cup of condensed milk.

a. How much condensed milk will Nina need? Show your calculation.
b. Measuring cups usually come in sets of $\frac{1}{4},\frac{1}{3},\frac{1}{2},\text{ and }1$ cup. Draw a diagram to show two different ways that Nina could measure the condensed milk needed for $4$ pans of fudge.

77. Portions. Don purchased a bulk package of candy that weighs $5$ pounds. He wants to sell the candy in little bags that hold $\frac{1}{4}$ pound. How many little bags of candy can he fill from the bulk package?

Solution

$20$ bags

78. Portions. Kristen has $\frac{3}{4}$ yards of ribbon that she wants to cut into $6$ equal parts to make hair ribbons for her daughter’s $6$ dolls. How long will each doll’s hair ribbon be?

## Exercises: Writing Exercises

Instructions: For questions 79-82, answer the given writing exercises.

79. Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into $6$ or $8$ slices. Would he prefer $3$ out of $6$ slices or $4$ out of $8$ slices? Rafael replied that since he wasn’t very hungry, he would prefer $3$ out of $6$ slices. Explain what is wrong with Rafael’s reasoning.

Solution

80. Give an example from everyday life that demonstrates how $\frac{1}{2}\cdot\frac{2}{3}$ is $\frac{1}{3}$.

81. Explain how you find the reciprocal of a fraction.

Solution

82. Explain how you find the reciprocal of a negative number.

## Exercises: Add Fractions with a Common Denominator

Instructions: For questions 83-92, add.

83. $\frac{6}{13}+\frac{5}{13}$

Solution

$\frac{11}{13}$

84. $\frac{4}{15}+\frac{7}{15}$

85. $\frac{x}{4}+\frac{3}{4}$

Solution

$\frac{x+3}{4}$

86. $\frac{8}{q}+\frac{6}{q}$

87. $-\frac{3}{16}+\left(-\frac{7}{16}\right)$

Solution

$-\frac{5}{8}$

88. $-\frac{5}{16}+\left(-\frac{9}{16}\right)$

89. $-\frac{8}{17}+\frac{15}{17}$

Solution

$\frac{7}{17}$

90. $-\frac{9}{19}+\frac{17}{19}$

91. $\frac{6}{13}+\left(-\frac{10}{13}\right)+\left(-\frac{12}{13}\right)$

Solution

$-\frac{16}{13}$

92. $\frac{5}{12}+\left(-\frac{7}{12}\right)+\left(-\frac{11}{12}\right)$

## Exercises: Subtract Fractions with a Common Denominator

Instructions: For questions 93-106, subtract.

93. $\frac{11}{15}-\frac{7}{15}$

Solution

$\frac{4}{15}$

94. $\frac{9}{13}-\frac{4}{13}$

95. $\frac{11}{12}-\frac{5}{12}$

Solution

$\frac{1}{2}$

96. $\frac{7}{12}-\frac{5}{12}$

97. $\frac{19}{21}-\frac{4}{21}$

Solution

$\frac{5}{7}$

98. $\frac{17}{21}-\frac{8}{21}$

99. $\frac{5y}{8}-\frac{7}{8}$

Solution

$\frac{5y-7}{8}$

100. $\frac{11z}{13}-\frac{8}{13}$

101. $-\frac{23}{u}-\frac{15}{u}$

Solution

$-\frac{38}{u}$

102. $-\frac{29}{v}-\frac{26}{v}$

103. $-\frac{3}{5}-\left(-\frac{4}{5}\right)$

Solution

$\frac{1}{5}$

104. $-\frac{3}{7}-\left(-\frac{5}{7}\right)$

105. $-\frac{7}{9}-\left(-\frac{5}{9}\right)$

Solution

$-\frac{2}{9}$

106. $-\frac{8}{11}-\left(-\frac{5}{11}\right)$

## Exercises: Mixed Practice

Instructions: For questions 107-114, simplify.

107. $-\frac{5}{18}\cdot\frac{9}{10}$

Solution

$-\frac{1}{4}$

108. $-\frac{3}{14}\cdot\frac{7}{12}$

109. $\frac{n}{5}-\frac{4}{5}$

Solution

$\frac{n-4}{5}$

110. $\frac{6}{11}-\frac{s}{11}$

111. $-\frac{7}{24}+\frac{2}{24}$

Solution

$-\frac{5}{24}$

112. $-\frac{5}{18}+\frac{1}{18}$

113. $\frac{8}{15}\div\frac{12}{5}$

Solution

$\frac{2}{9}$

114. $\frac{7}{12}\div\frac{9}{28}$

## Exercises: Add or Subtract Fractions with Different Denominators

Instructions: For questions 115-138, add or subtract.

115. $\frac{1}{2}+\frac{1}{7}$

Solution

$\frac{9}{14}$

116. $\frac{1}{3}+\frac{1}{8}$

117. $\frac{1}{3}-\left(-\frac{1}{9}\right)$

Solution

$\frac{4}{9}$

118. $\frac{1}{4}-\left(-\frac{1}{8}\right)$

119. $\frac{7}{12}+\frac{5}{8}$

Solution

$\frac{29}{24}$

120. $\frac{5}{12}+\frac{3}{8}$

121. $\frac{7}{12}-\frac{9}{16}$

Solution

$\frac{1}{48}$

122. $\frac{7}{16}-\frac{5}{12}$

123. $\frac{2}{3}-\frac{3}{8}$

Solution

$\frac{7}{24}$

124. $\frac{5}{6}-\frac{3}{4}$

125. $-\frac{11}{30}+\frac{27}{40}$

Solution

$\frac{37}{120}$

126. $-\frac{9}{20}+\frac{17}{30}$

127. $-\frac{13}{30}+\frac{25}{42}$

Solution

$\frac{17}{105}$

128. $-\frac{23}{30}+\frac{5}{48}$

129. $-\frac{39}{56}-\frac{22}{35}$

Solution

$-\frac{53}{40}$

130. $-\frac{33}{49}-\frac{18}{35}$

131. $-\frac{2}{3}-\left(-\frac{3}{4}\right)$

Solution

$\frac{1}{12}$

132. $-\frac{3}{4}-\left(-\frac{4}{5}\right)$

133. $1+\frac{7}{8}$

Solution

$\frac{15}{8}$

134. $1-\frac{3}{10}$

135. $\frac{x}{3}+\frac{1}{4}$

Solution

$\frac{4x+3}{12}$

136. $\frac{y}{2}+\frac{2}{3}$

137. $\frac{y}{4}-\frac{3}{5}$

Solution

$\frac{4y-12}{20}$

138. $\frac{x}{5}-\frac{1}{4}$

## Exercises: Mixed Practice

Instructions: For questions 139-152, simplify.

139.

a. $\frac{2}{3}+\frac{1}{6}$
b. $\frac{2}{3}\div\frac{1}{6}$

Solution

a. $\frac{5}{6}$
b. $4$

140.

a. $-\frac{2}{5}-\frac{1}{8}$
b. $-\frac{2}{5}\cdot\frac{1}{8}$

141.

a. $\frac{5n}{6}\div\frac{8}{15}$
b. $\frac{5n}{6}-\frac{8}{15}$

Solution

a. $\frac{25n}{16}$
b. $\frac{25n-16}{30}$

142.

a. $\frac{3a}{8}\div\frac{7}{12}$
b. $\frac{3a}{8}-\frac{7}{12}$

143. $-\frac{3}{8}\div\left(-\frac{3}{10}\right)$

Solution

$\frac{5}{4}$

144. $-\frac{5}{12}\div\left(-\frac{5}{9}\right)$

145. $-\frac{3}{8}+\frac{5}{12}$

Solution

$\frac{1}{24}$

146. $-\frac{1}{8}+\frac{7}{12}$

147. $\frac{5}{6}-\frac{1}{9}$

Solution

$\frac{13}{18}$

148. $\frac{5}{9}-\frac{1}{6}$

149. $-\frac{7}{15}-\frac{y}{4}$

Solution

$\frac{-28-15y}{60}$

150. $-\frac{3}{8}-\frac{x}{11}$

151. $\frac{11}{12a}\cdot\frac{9a}{16}$

Solution

$\frac{33}{64}$

152. $\frac{10y}{13}\cdot\frac{8}{15y}$

## Exercises: Use the Order of Operations to Simplify Complex Fractions

Instructions: For questions 153-174, simplify.

153. $\frac{{2}^{3}+{4}^{2}}{{\left(\frac{2}{3}\right)}^{2}}$

Solution

$54$

154. $\frac{{3}^{3}-{3}^{2}}{{\left(\frac{3}{4}\right)}^{2}}$

155. $\frac{{\left(\frac{3}{5}\right)}^{2}}{{\left(\frac{3}{7}\right)}^{2}}$

Solution

$\frac{49}{25}$

156. $\frac{{\left(\frac{3}{4}\right)}^{2}}{{\left(\frac{5}{8}\right)}^{2}}$

157. $\frac{2}{\frac{1}{3}+\frac{1}{5}}$

Solution

$\frac{15}{4}$

158. $\frac{5}{\frac{1}{4}+\frac{1}{3}}$

159. $\displaystyle\frac{\frac{7}{8}-\frac{2}{3}}{\frac{1}{2}+\frac{3}{8}}$

Solution

$\frac{5}{21}$

160. $\displaystyle\frac{\frac{3}{4}-\frac{3}{5}}{\frac{1}{4}+\frac{2}{5}}$

161. $\frac{1}{2}+\frac{2}{3}\cdot\frac{5}{12}$

Solution

$\frac{7}{9}$

162. $\frac{1}{3}+\frac{2}{5}\cdot\frac{3}{4}$

163. $1-\frac{3}{5}\div\frac{1}{10}$

Solution

$-5$

164. $1-\frac{5}{6}\div\frac{1}{12}$

165. $\frac{2}{3}+\frac{1}{6}+\frac{3}{4}$

Solution

$\frac{19}{12}$

166. $\frac{2}{3}+\frac{1}{4}+\frac{3}{5}$

167. $\frac{3}{8}-\frac{1}{6}+\frac{3}{4}$

Solution

$\frac{23}{24}$

168. $\frac{2}{5}+\frac{5}{8}-\frac{3}{4}$

169. $12\left(\frac{9}{20}-\frac{4}{15}\right)$

Solution

$\frac{11}{5}$

170. $8\left(\frac{15}{16}-\frac{5}{6}\right)$

171. $\displaystyle\frac{\frac{5}{8}+\frac{1}{6}}{\frac{19}{24}}$

Solution

$1$

172. $\displaystyle\frac{\frac{1}{6}+\frac{3}{10}}{\frac{14}{30}}$

173. $\left(\frac{5}{9}+\frac{1}{6}\right)\div\left(\frac{2}{3}-\frac{1}{2}\right)$

Solution

$\frac{13}{3}$

174. $\left(\frac{3}{4}+\frac{1}{6}\right)\div\left(\frac{5}{8}-\frac{1}{3}\right)$

## Exercises: Evaluate Variable Expressions with Fractions

Instructions: For questions 175-184, evaluate.

175. $x+\left(-\frac{5}{6}\right)$ when

a. $x=\frac{1}{3}$
b. $x=-\frac{1}{6}$

Solution

a. $-\frac{1}{2}$
b. $-1$

176. $x+\left(-\frac{11}{12}\right)$ when

a. $x=\frac{11}{12}$
b. $x=\frac{3}{4}$

177. $x-\frac{2}{5}$ when

a. $x=\frac{3}{5}$
b. $x=-\frac{3}{5}$

Solution

a. $\frac{1}{5}$
b. $-1$

178. $x-\frac{1}{3}$ when

a. $x=\frac{2}{3}$
b. $x=-\frac{2}{3}$

179. $\frac{7}{10}-w$ when

a. $w=\frac{1}{2}$
b. $w=-\frac{1}{2}$

Solution

a. $\frac{1}{5}$
b. $\frac{6}{5}$

180. $\frac{5}{12}-w$ when

a. $w=\frac{1}{4}$
b. $w=-\frac{1}{4}$

181. $2{x}^{2}{y}^{3}$ when $x=-\frac{2}{3}$ and $y=-\frac{1}{2}$

Solution

$-\frac{1}{9}$

182. $8{u}^{2}{v}^{3}$ when $u=-\frac{3}{4}$ and $v=-\frac{1}{2}$

183. $\frac{a+b}{a-b}$ when $a=-3,b=8$

Solution

$-\frac{5}{11}$

184. $\frac{r-s}{r+s}$ when $r=10,s=-5$

## Exercises: Everyday Math

Instructions: For questions 185-186, answer the given everyday math word problems.

185. Decorating. Laronda is making covers for the throw pillows on her sofa. For each pillow cover, she needs $\frac{1}{2}$ yard of print fabric and $\frac{3}{8}$ yard of solid fabric. What is the total amount of fabric Laronda needs for each pillow cover?

Solution

$\frac{7}{8}$ yard

186. Baking. Vanessa is baking chocolate chip cookies and oatmeal cookies. She needs $\frac{1}{2}$ cup of sugar for the chocolate chip cookies and $\frac{1}{4}$ of sugar for the oatmeal cookies. How much sugar does she need altogether?

## Exercises: Writing Exercises

Instructions: For questions 187-188, answer the given writing exercises.

187. Why do you need a common denominator to add or subtract fractions? Explain.

Solution

188. How do you find the LCD of $2$ fractions?